Can this differential equation be solved using separation of variables? $\dfrac{dy}{dx}=\dfrac{3}{x^2y-8y}$ Choose 1 answer: Choose 1 answer: (Choice A) A Yes (Choice B) B No
Explanation: For an equation to be solvable using separation of variables, we need to be able to bring it to the form $\dfrac{dy}{dx}=f(x)g(y)$. In this form, $f(x)$ doesn't include $y$ and $g(y)$ doesn't include $x$. Notice that we must multiply, not add, $f(x)$ and $g(y)$. First, let's factor $\dfrac{3}{x^2y-8y}$ as $\left(\dfrac{3}{x^2-8}\right)\left(\dfrac{1}{y}\right)$. Now, let $f(x)=\dfrac{3}{x^2-8}$ and $g(y)=\dfrac{1}{y}$. Our equation is indeed in the form $\dfrac{dy}{dx}=f(x)g(y)$. Yes, the equation can be solved using separation of variables.